Problem 44831. Composing relative poses in 2D: problem 2

We consider a world reference frame denoted by {0} which has its x-axis pointing east and its y-axis pointing north.

There is a playing field with a reference frame denoted by {F}. The rigid-body displacement from {O} to {F} is given by the homogenous transformation TF .

There is a robot with an attached body-fixed coordinate frame {B} whose origin is in the centre of the robot, and whose x-axis points in the robot's forward direction. The rigid-body displacement displacement of {B} relative to {F} is given by the homogenous transformation TB .

The robot carries a laser scanner which reports the range and bearing of landmarks with respect to its coordinate frame {S} with zero bearing angle corresponding to its x-axis. The scanner is displaced from the centre of the robot and the displacement of frame {S} relative to {B} is given by the homogenous transformation TS .

There is a landmark with position described by a coordinate vector P with respect to the world coordinate frame {O}.

What is the range and bearing of the landmark as observed by the laser scanner?